Collateral Constraints and Multiplicity · Collateral Constraints and Multiplicity Jess Benhabib Pengfei Wang New York University April 17, 2013 Jess Benhabib Pengfei Wang (New York

Collateral Constraints and Multiplicity

Jess Benhabib Pengfei Wang

New York University

April 17, 2013

Jess Benhabib Pengfei Wang (New York University)Collateral Constraints and Multiplicity April 17, 2013 1 / 44

Introduction

Firms and businesses face borrowing costs to finance their workingcapital that depends on the collateral value of their assets and output.Any positive shock that appreciates the value of a firm’s collateral willdecrease the cost of external finance, increase profitability and amplifythe effect of the initial shock.

This mechanism suggests the possibility of self-fulfilling multipleequilibria: Optimistic expectations of higher output may well leadincreased lending to financially constrained firms.


Mechanism

Even though our model has no increasing returns in production, therelaxation of the borrowing constraint implies that unit marginal costscan increase with output as firms compete hire more labor andcapital. In such a case markups can become countercyclical andfactor returns can increase suffi ciently so that the expectation ofhigher output can become self-fulfilling.

The purpose of this paper is to show that multiple equilibria andindeterminacies can easily arise in a simple financial accelerator modelfor realistic parameter calibrations, and that it can reasonably matchsome of the quantitative features of economic data.


We introduce borrowing constraints into an otherwise standardDixit-Stiglitz monopoly competition model. Firms rent capital andhire labor in the competitive markets to produce differentiatedintermediate goods.

The firms however may default on their promise or contract to repaytheir debt. We assume therefore that firms face borrowing constraintsto finance their working capital, determined by the fraction of firmrevenue and assets that the creditors can recover, minus some fixedcollection costs. This constrains the output as well as the unitmarginal costs of firms.


Given the fixed collection costs however, if households expect a higherequilibrium output, they will be willing to increase their lending tofirms, even if the marginal costs of firms increase and their markupsdecline as they compete for additional labor and capital.

At the new equilibrium both output and factor returns will be higher.Despite the income effects on labor supply, the increase in wagesassociated with lower markups will allow employment and output toincrease, so the optimistic expectations of higher output will befulfilled.


Final good

Yt =[∫

Yσ−1

σt (i)di

] σσ−1,

The final goods producer solves

maxyt (i )

[∫Y

σ−1σ

t (i)di] σ

σ−1−∫Pt (i)Yt (i)di .

where Pt (i) the price of the i’th intermediate good.The first-order conditions lead to the following inverse demandfunctions for intermediate goods:

Pt (i) = Y− 1

σt (i)Y

1σt ,

where the aggregate price index is

1 =[∫

P1−σt (i)di

] 11−σ

.


The Financial Constraint.

We assume that in the beginning of period, the ith intermediategoods firm decides to rent capital Kt (i) from the households and hirelabor Nt (i). The firm promises to pay wtNt (i) + rtKt (i) ≡ bt (i) tothe households.

However the firm may default on their contract or promise. Weassume that if the firm does not pay its debt bt (i), the householdscan recover a fraction ξ < 1 of the firm’s revenue Pt (i)Yt (i) byincurring a liquidation cost f .

One possibility is that the firm must pay the wages to labor asproduction takes place , and that creditors can always redeem thephysical capital, but that the interest on borrowing may not be fullyrecoverable.


So if the household can recover ξPt (i)Yt (i)− f , they will lend to thefirm only if ξPt (i)Yt (i)− f will at least cover the wage bill plusprincipal and interest.

Knowing that the household can not recover more thanξPt (i)Yt (i)− f , the firm will also has no incentive to repay morethan ξPt (i)Yt (i)− f . The incentive-compatiblity constraint for thefirm then is:

Pt (i)Yt (i)− [wtNt (i) + rtKt (i)]≥ Pt (i)Yt (i)− [ξPt (i)Yt (i)− f ] ,

orwtNt (i) + rtKt (i) ≤ ξPt (i)Yt (i)− f .


Cost Minimization

Min rtKt + wtNt S .T . AK at N1−αt ≥ Yt

Max L = rtKt + wtNt − φt(Yt − A K at N1−α

t

)FOC:

rt = αφtA Ka−1t N1−α

t ; wt = (1− α) φtA Kat N−αt

where φt is the Lagrange multiplier interpreted as marginal cost, themarginal effect of relaxing the income constraint on marginal costs. Then( rt

α

)α=(φtA K

a−1t N1−α

t

)α;

(wt1− α

)1−α

=(φtA K

at N−αt

)1−α

Mulltiplying:( rtα

)α(wt1− α

)1−α

= φt

(A K α(α−1)+α(1−α)

t Nα(1−α)−α(1−α)t

)= Aφt

φt = A−1( rt

α

)α(wt1− α

)1−α


After substituting Pt (i), the profit maximization for the i ′th firmbecomes

maxYt (i )

Y1− 1

σt (i)Y

1σt − φtYt (i),

subject to

φtYt (i) + f ≤ ξY1− 1

σt (i)Y

1σt .

Given wt , rt , final output Yt , and the borrowing constraint, thefeasible choice of Yt (i) can be illustrated by shaded area of Figure 1.


Figure 1. The Credit Constraints and Feasible Ouptut Choice.


Households

∫ ∞

0[logCt − ψ

N1+χt

1+ χ]e−ρtdt,

subject to

Kt = rtetKt − δ(et )Kt + wtNt − Ct +Πt ,

For simplicity we assume that the households choose the capacityutilization rate et . A higher et implies that the capital is moreintensively utilized, at the cost of faster depreciation, so that δ(et ) isconvex increasing function.


The first-order conditions for the consumer’s optimization problem aregiven by equations:

CtCt= rtet − ρ− δ(et ),

rt = δ′(et ),

andψNχ

t =1Ctwt .


The equilibrium in the economy is a collection of price processes{wt , rt ,Pt (i)} and quantities {Kt (i),Nt (i),Yt (i),Yt ,Kt ,Nt , et ,Πt},such that given the prices and the aggregate Πt , the householdschoose Kt and Nt to maximize their utility;

Given Pt (i), the final goods firm chooses {Yt (i)} to maximize itsprofits

Given wt , rt , and the financial constraint (the intermediate goodsmaximizes its profit by choosing Kt (i) and Nt (i); and all marketsclear.

Since firms are symmetric, we have Kt (i) = Kt , Nt (i) = Nt ,Pt (i) = 1, Yt (i) = Yt and Πt (i) = Πt = Yt − wtNt − rtKt .


The budget constraint becomes

Kt = Yt − Ct − δKt .

The wage wt and the interest rate rt are

wt = (1− α)φtYtNt,

and

rt = αφtYtKt.


Lemma: If ξ(1− 1σ ) < φt < 1− 1

σ , then the final constraint binds;that is

φtYt (i) + f ≤ ξY1− 1

σt (i)Y

1σt .

Why? Marginal revenue = 1− 1σ > Marginal cost = φt > Marginal

Revenue in case of default = ξ(1− 1

σ

)First inequality says firms want to borrow more to expand output,second inequality says they cannot because what the household canrecover in case of default does not cover marginal cost.

Using the fact Yt (i) = Yt , the constraint implies

φt = ξ − fYt


To summarize, the following system of equations fully characterize theequilibrium

CtCt

= φt

αYtKt− ρ− δ(et ),

Kt = Yt − δ(et )Kt − Ct ,

ψNχt =

1Ct

φt(1− α)Yt

Nt,

Yt = A(etKt )αN1−αt ,

φt

αYtetKt

= δ′(et )

φt = ξ − fYt,

subject to the constraint ξ(1− 1σ ) < φt < 1− 1

σ .

Let the depreciation function be given by δ(et ) = δ0e1+µt1+µ .We then

have

φαYeK

= δ′(e) = δ0eµ


We first solve deterministic steady state. We normalize δ0 such that e = 1.

δ(e) =1

1+ µφ

αYK

K =µ

1+ µ

φα

ρY

e =[(1+ µ)ρ

µδ0

] 11+µ

CY

= 1− δ(e)KY

= 1− ρ

µ

µ

1+ µ

φα

ρ

= 1− φα

1+ µ

N =

[φ(1− α)

CY

1ψ

] 11+χ


Y =(

µ

1+ µ

φα

ρ

) α1−α

φ(1− α)

1− φα1+µ

1ψ

11+χ

≡ Y (φ)

Finally from the definition of φ = ξ − fY , we have

f = (ξ − φ)Y (φ) ≡ Ψ(φ)

to determine the steady-state value of φ. For the existence of a steadystate however we will need to assume ξ(1− 1

σ ) < φ < 1− 1σ . See

diagram: f cannot be too high, or alternatively constrain φ.


ct = ρ[Yt − Kt + φt ]

kt =(1+ µ)δ

αφ(Yt − Kt )−

((1+ µ)δ

αφ− δ

)(Ct − Kt )

−δ(Yt + φt − Kt )χNt = φt + Yt − Nt − CtYt = α(Kt + et ) + (1− α)Nt

et =1

1+ µ(φ

t+ Yt − Kt )

φt =f /Y

ξ − f /YYt ≡ γYt

where we use δ = ρµ .


[ktct

]= J

[KtCt

](1)

Using the factor ρ = δµ, we have

J = δ

[(1+µ)

αφ λ1 − (1+ γ)λ1(1+µ)

αφ (λ2 − 1) + 1− (1+ γ)λ2µ [(1+ γ)λ1 − 1] µ(1+ γ)λ2

]


Proposition: Let γ and φ satisfy the following two constraints

(1+ γ) >(1+ µ) (1+ χ)

α(1+ χ) + (1+ µ)(1− α)

and

1+ γ < min(1+ µ

α,

(1+µ

φ

)(1+ χ)

α(1+ χ) + (1+ µ)(1− α),

(1− α)(1+ χ)

(1+ µ)(1− α) 11+µ−αφ + (1+ χ)α

+ 1)

ThenTrace (J) < 0, det(J) > 0


To gain intuition for self-fulfilling expectations of higher output andhigher factor rewards, we first focus on labor demand and supplycurves incorporating the equilibrium effects of the borrowingconstraint on marginal costs and markups.

The labor demand curve is given by

wt = (1+ γ)Yt − Nt (2)

wt =αν

1+ ν− (1+ γ)αKt +

[(1+ γ)(1+ ν)(1− α)

1+ ν− (1+ γ)α− 1]Nt(3)

and the labor supply curve in the economy is

wt = Ct + χNt (4)


This slope of the labor market demand curve is positive and steeperthan the labor supply curve under the condition

(1+ γ) >(1+ µ) (1+ χ)

α(1+ χ) + (1+ µ)(1− α)

.

Unlike earlier works, our model has no increasing returns in theproduction technology. Instead indeterminacy arises from theborrowing constraints and their indirect effects on marginal coststhrough wages and the rental rate on capital. If households expect ahigher equilibrium output, they will be willing to increase their lendingto firms.


Given the positive fixed collection costs f , an expected increase inoutput levels relaxes the borrowing constraint so that unit marginalcosts of firms, φt = ξ − f

Yt, can rise and markups can decline.

This implies that as firms compete for inputs, factor rewards will alsowith increase with Yt . The labor demand curve incorporating thesegeneral equilibrium effects on marginal costs can now be positivelysloped and steeper than the labor supply curve.

Normally, higher output levels tend to increase the demand for leisure,so barring inferiorities in preferences, the higher demand for labor willbe contained by the income effect on labor supply.


However if the labor demand slopes up more steeply than laborsupply, employment will increase robustly as the labor supply curveshifts to the left with income effects.

The rise in labor hours as well as the accumulation of capital will raiseoutput, so that optimistic output expectations of households will beself-fulfilling.

We can show that these indeterminacy results can hold even in theabsence of variable capacity utilization, but we include it into ourmodel to improve calibration results in the next section.


Figure 2 illustrates the combinations of f and ξ which may yieldindeterminacy with the other parameters set at µ = 0.3, α = 1

3 ,ρ = 0.01. The shaded areas are the feasible ξ and f that support asteady state equilibrium with ξ σ−1

σ < φ < σ−1σ .

Figure 2. Parameter Spaces for Indeterminacy.Jess Benhabib Pengfei Wang (New York University)Collateral Constraints and Multiplicity April 17, 2013 27 / 44

Calibrations

We write the model in discrete time and solve it by log-linearizing theequations that characterize the equilibrium around the steady state. Weadopt standard parameterization: β = 1

1+ρ = 0.99, α = 1/3,δ = 0.033, σ = 10 and µ = 0.3. We set ξ = 0.9768, f = 0.1908 and fixthe productivity level at A = 1. These parameter values imply steady statevalues φ = 0.88 and γ = f /Y

ξ−f /Y = 0.11.


We begin without fundamental shocks. In the case of indeterminacy, themodel’s solution takes the form(

Kt+1Ct+1

)= M

(KtCt

)+

(0

εt+1

)where M is a two by two matrix and εt+1 = Ct+1 − Et Ct+1 is the sunspotshock. The remaining variables can be written as functions of Kt and Ct :

YtItNtet

= H(KtCt

)


Figre 3. Impulse Responses to a Consumption Shock.


From Figure 3 we see that output, investment, consumption andhours comove.

The impulse responses also demonstrate that labor is slightly morevolatile than output, an important feature of the data that thestandard RBC model has diffi culty explaining with a T.F.P shock.

The impulse responses also show cycles in output, investment,consumption and hours, so the model has the potential to explain theboom-bust patterns observed in many episodes in data.


However, as in the models with increasing returns to scale, theextremely large impact of autonomous consumption on output andinvestment seems empirically unjustified.

On the impact period, one percentage increase consumption leads to27 percent increase in output and 116 percent increase in investment..


This volatile response of output and investment can be understood bystudying the effect of consumption on labor. Equating the labor demand(2) and labor supply (4) we have

Nt =1

(1+γ)(1+µ)(1−α)1+µ−(1+γ)α

− 1− χCt . (5)

where (1+γ)(1+µ)(1−α)1+µ−(1+γ)α

− 1 is the slope of the labor demand curve and χ isthe slope of labor supply curve. When these two slopes are close, a onepercentage point of autonomous consumption increase can lead to a hugeincrease in labor and hence output. Denote s as the steady stateinvestment to income ratio. Then from the resource constraint,

sIt + (1− s)Ct = Yt , (6)

so it is clear that smooth consumption plus volatile income will makeinvestment even more volatile as s << 1. In the current calibrations = 0.23. So it implies that response of investment at impact will beabout 4.4 times that of output.


Table 1: Sample and Model Moments

US Sample Modelvar σX /σY corrXY corrXtXt−1 σX /σY corrXY corrXtXt−1Y 1.00 1.00 0.87 1.00 1.00 0.91N 1.01 0.88 0.92 1.08 0.99 0.91C 0.52 0.83 0.90 0.07 0.48 0.97I 3.33 0.92 0.92 4.31 0.99 0.91φ 0.32 0.16 0.70 0.11 1.00 0.91

Note: Variables (Y ,N,C , I , φ) stands for output, labor(hours), consumption, investment and marginal cost respectively.The marginal cost in the data can be computed viaφ = labor share

1−α . σX /σY is the relatively standard deviation ofvariable X to output, corr(X ,Y ) computes the correlationbetween X and output and corr(Xt ,Xt−1) compute thefirst-order autocorrelation of Xt .


To better match the relative volatility of consumption and output we nowintroduce a TFP shock into the model. We assume that the technologylevel in the economy follows an AR(1) process

At+1 = ρaAt + σaεat+1 (7)

Following Benhabib and Wen (2004), we assume the sunspots shocks andtechnology shocks are correlated. Following King and Rebelo (1999) , weassume ρa = 0.98. We assume the technology shock εat and sunspotsshocks εt are perfectly correlated and the relative volatility of sunspots andtechnology shocks is set to σs/σε = 1.5. These bring the relative volatilityof consumption closer to data. The moments with correlated TFP shocksand sunspots shocks are in Table 2.


Table 2: Moments with correlated TFP and Sunspot Shocks

Model with Correlated Shocks The RBC Modelvar σX /σY corrXY corrXtXt−1 σX /σY corrXY corrXtXt−1Y 1.00 1.00 0.98 1.00 1.00 0.95N 0.96 0.95 0.99 0.53 0.73 0.90C 0.37 0.55 0.99 0.62 0.86 0.99I 3.38 0.96 0.98 2.65 0.88 0.91φ 0.11 1.00 0.98 0 N.A N.A

The RBC model refers to f = 0, so γ = 0, in which we haveφ = ξ is a constant. We select the parameter values in a waysuch that these two models have the same steady state. For theRBC model, we use TFP shocks with ρa = 0.98 as the onlydriving forces.


Hump-Shaped output Dynamics

We illustrate how our indeterminacy model can also predict some ofthe aspects of actual fluctuations that standard RBC models cannotexplain, such as the hump-shaped, trend reverting impulse response ofoutput to transitory demand shocks and the substantial serialcorrelation in the output growth rate in data (see Cogley andNason(1995)).

Since there is significant empirical evidence favoring demand shocksas a main source of business cycle, (e.g., see Blanchard andQuah(1989), Waston (1993), Cogley and Nason (1995) and Benhabiband Wen (2004), it is important to examine whether demand shockscan generate persistent business cycles.


We consider two types of demand shocks as in Benhabib and Wen(2004): government spending shocks and preference shocks. Withpreference shocks the period-by-period utility function changes to

U = exp(∆t ) logCt − ψN 1+χt1+χ . We assume that the preference shocks

∆t follows an AR(1) process, namely ∆t = ρ∆∆t−1 + ε∆t . Withgovernment spending, Gt , in period t, the resource constraint changesto Kt = Yt − δ(et )Kt − Ct − Gt . We assumelog(Gt ) = ρg log(Gt−1) + εgt . We choose ρg = ρ∆ = 0.90 as inBenhabib and Wen (2004).


To highlight the effect of indeterminacy on the propagationmechanism of RBC models, we graph the impulse responses to apersistent government spending shock with and withoutindeterminacy in Figure 4. Figure 5 graphs the impulse response ofthe model to a persistent preference shock. For the model withoutindeterminacy we set f = 0 and reset ξ = 0.88 such that the modelwith and without indeterminacy have the same steady state.Several features of Figure 4 deserves particular mention.First, in the case f = 0, we have marginal cost φt = ξ is a constant.Hence the impulse responses of our model with financial constraintsresemble these of a standard RBC model. Figure 4 and Figure 5 showthat some diffi culties of the standard RBC model in generatingbusiness cycle fluctuations. Figure 4 shows that consumption andinvestment move against each other after a positive governmentspending shock. An increase in government spending generates anegative wealth effect, which reduces both consumption and leisure.The decrease in leisure leads to an increase in output. An increase inoutput together with a decrease in consumption imply thatinvestment has to increase.


Second, even though the model generates comovement withoutindeterminacy under persistent preference shocks, the responses ofoutput to such demand shocks is monotonic. Neither governmentspending shocks nor preference shocks can generate the hump-shapedoutput dynamics observed in the data. And this monotonic andpersistent output responses to demand shocks mostly come from thepersistence of shocks, not from an inner propagation mechanism. Ifone reduces the persistence of the shocks, the persistence of outputresponses will reduced accordingly.


Third, when the model is indeterminate, the responses of output toboth the government spending shocks and the preference shocks aredramatically changed.

Figure 4 and Figure 5 clearly show persistent and hump-shapedresponses of output to both shocks. In addition, this persistentresponse of output is not due to the persistence in shocks. As Figure3 has already demonstrated, the model with indeterminacy cangenerate persistent fluctuations even under i.i.d shocks.

Figure 4 and Figure 5 again highlight the similarity of ourindeterminacy model with those based on increasing return to scale,so it has a similar ability to explain other puzzles. For example,Benhabib and Wen (2004) demonstrated that their indeterminacymodel based on increasing returns to scale can explain theforcecastable-movement puzzle as pointed out by Rotemberg andWoodford (1996), namely that they are highly forecastable andcomove.


Figure 4. Impulse response to a government shock.

Solid lines are responses under determinacy (f = 0) and dashed lines areresponses under indeterminacy.


Figure 5. Impulse responses to a preference shock.

Solid lines are responses under determinacy (f = 0) and dashed lines areresponses under indeterminacy.


We conclude that borrowing or collateral constraints can be a source ofself-fulfilling fluctuations in economies that have no increasing returns toscale in production. Expectations of higher output can relax borrowingconstraints, and firms can expand their output by bidding up factor pricesand eliciting a labor supply response that allows the initial expectation tobe fulfilled. The parameter ranges and markups where self-fulfillingexpectations can occur are within realistic ranges and compatible with thedata. Simulating our data we obtain moments and impulse responses thatcan reasonably match the US macroeconomic data.


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Collateral Constraints and Multiplicity · Collateral Constraints and Multiplicity Jess Benhabib Pengfei Wang New York University April 17, 2013 Jess Benhabib Pengfei Wang (New York